Cosmological Einstein-Skyrme solutions with nonvanishing topological charge

Time-dependent analytic solutions of the Einstein-Skyrme system - gravitating Skyrmions - with topological charge are analyzed in detail. In particular, the question of whether these Skyrmions reach a spherically symmetric configuration for t→+∞ is discussed. It is shown that there is a static, spherically symmetric solution described by the Ermakov-Pinney system, which is fully integrable by algebraic methods. For Λ>0, this spherically symmetric solution is found to be in a "neutral equilibrium" under small deformations, in the sense that under a small squashing it would neither blow up nor disappear after a long time, but it would remain finite forever (plastic deformation). Thus, in a sense, the coupling with Einstein gravity spontaneously breaks the spherical symmetry of the solution. However, in spite of the lack of isotropy, for t→∞ (and Λ>0), the spacetime is locally flat and the anisotropy of the Skyrmion only reflects the squashing of spacetime.